Model comparison using rjmcmc & gamma prior on B
Bayes Factor of compared models determined
Contents
- Part I: RJMCMC - \(exp(1)\) prior on \(\beta\)
- Data
- Data simulated using the Super Spreading Events model \((\alpha, \beta, \gamma)\). Four datasets simulated - small & large outbreaks.
- Data simulated using the Baseline model of a regular events spreading model \((\alpha)\). Two datasets simulated.
- RJMCMC
- RJMCMC implemented, jumping between the SSE and baseline model.
- Bayes Factor between the two models calculated
- Part II: RJMCMC - \(\Gamma(2, 1/2)\) prior on \(\beta\)
- The steps are repeated as in part I with the exception of a \(\Gamma(2, 1/2)\) prior on \(\beta\)
- Identical datasets are used
Background
RJMCMC was implemented whereby the algorithm jumped between the Baseline model \(M_1\) of an epidemic with regular spreading events \((\alpha)\) and the Super Spreading Events (SSE) model \((\alpha, \beta, \gamma)\) which has both regular spreading events with rate \(\alpha\) and super spreading events with rate \(\beta\) and multiplicative factor \(\gamma\).
Bayes Factor
Bayesian model comparison is a method of model selection based on Bayes factors. The aim of the Bayes factor is to quantify the support for one model over another, e.g model \(M_1\) over model \(M_2\). The Bayes Factor BF is as follows;
\[ BF = \dfrac{P(D|M_1)}{P(D|M_2)} = \dfrac{\dfrac{P(M_1|D)P(D)}{P(M_1)}}{\dfrac{P(M_2|D)P(D)}{P(M_2)}} = \dfrac{P(M_1|D)}{P(M_2 | D)}
\]
when \(P(M_1) == P(M_2)\), otherwise
\[ BF = \dfrac{P(D|M_1)}{P(D|M_2)} = \dfrac{\dfrac{P(M_1|D)}{P(M_1)}}{\dfrac{P(M_2|D)}{P(M_2)}} = \dfrac{P(M_1|D)}{P(M_1)} \cdot \dfrac{P(M_2)}{P(M_2|D)}
\]
where \(P(D|M_1)\) is the model evidence, specifically the marginal likelihood integrand;
\[ P(D|M_1) = \int P(D \hspace{1 mm}|\hspace{1 mm} M_1, \theta) \hspace{1 mm} P(\hspace{1 mm}\theta \hspace{1 mm}| M_1) \hspace{1 mm}d \theta\] and the first term in the integrand \(P(D \hspace{1 mm}|\hspace{1 mm} M_1, \theta)\) is the likelihood and the second term \(P(\hspace{1 mm}\theta \hspace{1 mm}| M_1)\) is the prior on the model parameter \(\theta\).
Interpretation of the Bayes Factor results
A Bayes Factor > 1 signifies that \(M_1\) is more strongly supported by the data under consideration than \(M_2\). Harold Jefferys gave a scale of interpretation of the Bayes Factor;
| Bayes Factor | Bayes Factor equivalence | Evidence Strength |
|---|---|---|
| < 10^0 | < 1 | Negative (supports M_2) |
| [10^0, 10^1/2] | [1, 3.16] | Weak evidence |
| [10^1/2, 10^1] | [3.16, 10] | Substantial |
| [10^1, 10^3/2] | [10, 31.62] | Strong |
| [10^3/2, 10^2] | [31.62, 100] | Very strong |
| > 10^2 | > 100 | Decisive |
Part I: RJMCMC between Base model & SSE Model - exp(1) prior used for beta
An \(exp\hspace{1mm}(\beta; \hspace{1mm}1)\) has density;
Model Comparison - SSE data I
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 1 10000 0.8 0.31 0.1 0.01 10 0.19 1.8 0.25 70.85 86.68
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 983 12.7 144 99.56 1129 876
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 258 77.25 875 181 82.86 0.8866
## beta_pc_non_0 bf
## 1 0.1134 7.818
Model Comparison - SSE data II
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 1 10000 0.8 0.63 0.1 0.07 10 5.49 1.8 1.06 2.16 6.28
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 628 2.56 256 6.37 637 0
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 9999 0 0 0 NaN 0
## beta_pc_non_0 bf
## 1 1 0
Model Comparison - SSE data III
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 2 10000 0.8 0.45 0.1 0.16 10 6.62 1.8 1.46 4.71 5.18
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 518 1.91 191 5.42 542 0
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 9999 0 0 0 NaN 0
## beta_pc_non_0 bf
## 1 1 0
Model Comparison - SSE data IV
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 3 10000 0.8 0.77 0.1 0.02 10 0.26 1.8 0.8 10.32 84.7
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 1306 14.59 225 93 1434 1464
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 78 94.94 1464 1735 45.76 0.8457
## beta_pc_non_0 bf
## 1 0.1543 5.481
Model Comparison - BASE Data I
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 1 10000 0.8 1.75 0.1 0.02 10 0.18 1.8 1.77 3.63 53.78
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 675 8.69 109 43.35 544 1244
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 11 99.12 1243 4288 22.47 0.8745
## beta_pc_non_0 bf
## 1 0.1255 6.968
Model Comparison - BASE Data II
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 2 10000 0.8 1.74 0.1 0.08 10 0.43 1.8 1.85 9.88 62.6
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 1734 9.31 258 39.93 1106 2584
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 186 93.29 2584 1964 56.82 0.7229
## beta_pc_non_0 bf
## 1 0.2771 2.609Part 2: Gamma Prior on Beta
A gamma prior, \(\Gamma(\beta; k, \theta)\) on beta was also trialed whereby \(k\) determines the shape of the distribution and \(\theta\) governs the scale. The gamma distribution function is as follows;
\[ \Gamma(\beta; k, \theta) = \dfrac{1}{\Gamma(k) \cdot \theta^k}\cdot \beta^{(k-1)} \cdot e^{\dfrac{-\beta}{\theta}} \]
A range of gamma priors on beta are used including a \(\Gamma\hspace{1mm}(\beta; \hspace{1mm}2, \hspace{1mm} 2.5)\), \(\Gamma\hspace{1mm}(\beta; \hspace{1mm}2.5, \hspace{1mm} 2)\) and a \(\Gamma \hspace{1mm}(\beta; \hspace{1mm} 3, \hspace{1mm} 2)\) Each have a mean of \(k \cdot \theta\). A \(\Gamma\hspace{1mm}(\beta; \hspace{1mm}2, \hspace{1mm} 2.5)\), is as follows;
A \(\Gamma\hspace{1mm}(\beta; \hspace{1mm}3, \hspace{1mm} 2)\);
And a \(\Gamma\hspace{1mm}(\beta; \hspace{1mm}3, \hspace{1mm} 3)\);
And a \(\Gamma\hspace{1mm}(\beta; \hspace{1mm}4, \hspace{1mm} 4)\);
In the Metropolis acceptance step, the logs of all quantities are determined and evaluating \(log\hspace{1mm}( \Gamma\hspace{1mm}(\beta; \hspace{1mm} k, \hspace{1mm} \theta))\) gives;
\[log\bigg( \dfrac{1}{\Gamma(k) \cdot \theta^k}\cdot \beta^{(k-1)} \cdot e^{\dfrac{-\beta}{\theta}} \bigg)\]
\[ = \dfrac{1}{log\Gamma(k)\cdot klog(\theta)} \cdot (k-1) \cdot log(\beta) \cdot \dfrac{-\beta}{\theta} \]
Gamma Prior \(\Gamma\)(\(\beta\); 2, 2.5)
Model Comparison - SSE data I
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 1 10000 0.8 0.48 0.1 0.09 10 307.09 1.8 25.42 78.43 0.01
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 1 99.5 9856 0.03 3 38
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 9868 0.38 38 4 90.48 0.0093
## beta_pc_non_0 bf
## 1 0.9907 0.009
Model Comparison - SSE data II
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 1 10000 0.8 0.58 0.1 0.1 10 19.46 1.8 2.53 42.64 0
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 0 60.29 6028 0 0 0
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 9999 0 0 0 NaN 0
## beta_pc_non_0 bf
## 1 1 0
Model Comparison - SSE data III
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 2 10000 0.8 0.62 0.1 0.1 10 13.22 1.8 1.94 32.09 0
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 0 15.51 1551 0 0 0
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 9999 0 0 0 NaN 0
## beta_pc_non_0 bf
## 1 1 0
Model Comparison - SSE data IV
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 3 10000 0.8 1.02 0.1 0.08 10 3.8 1.8 1.52 22.89 0.18
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 6 52.78 1729 1.01 33 1938
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 1338 59.16 1938 1113 63.52 0.6723
## beta_pc_non_0 bf
## 1 0.3277 2.052
Model Comparison - BASE Data I
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 1 10000 0.8 1.77 0.1 0.02 10 0.18 1.8 1.79 3.38 0.08
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 1 18.77 234 0.56 7 1246
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 1 99.92 1246 4299 22.47 0.8752
## beta_pc_non_0 bf
## 1 0.1248 7.013
Model Comparison - BASE Data II
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 2 10000 0.8 1.74 0.1 0.09 10 0.48 1.8 1.87 9.94 0.28
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 8 16.86 488 2.8 81 2619
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 275 90.5 2619 1885 58.15 0.7105
## beta_pc_non_0 bf
## 1 0.2895 2.454
Gamma Prior \(\Gamma(\beta; 3, 2)\)
Model Comparison - SSE data I
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 1 10000 0.8 0.54 0.1 0.08 10 9 1.8 2.18 79.71 89.08
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 3849 83.89 3625 87.06 3762 1285
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 3036 29.74 1285 352 78.5 0.5678
## beta_pc_non_0 bf
## 1 0.4322 1.314
Model Comparison - SSE data II
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 1 10000 0.8 0.63 0.1 0.09 10 46.62 1.8 3.41 36.21 31.74
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 3174 61.09 6108 53.19 5318 0
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 9999 0 0 0 NaN 0
## beta_pc_non_0 bf
## 1 1 0
Model Comparison - SSE data III
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 2 10000 0.8 0.42 0.1 0.19 10 10.81 1.8 2.32 15.31 9.4
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 940 9.34 934 13.58 1358 0
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 9999 0 0 0 NaN 0
## beta_pc_non_0 bf
## 1 1 0
Model Comparison - SSE data IV
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 3 10000 0.8 0.95 0.1 0.05 10 0.82 1.8 1.06 17.04 82.39
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 2045 48.23 1197 81.91 2033 1766
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 716 71.15 1765 1398 55.8 0.7518
## beta_pc_non_0 bf
## 1 0.2482 3.029
Model Comparison - BASE Data I
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 1 10000 0.8 1.77 0.1 0.02 10 0.19 1.8 1.79 3.74 53.91
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 676 17.38 218 36.44 457 1238
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 16 98.72 1237 4343 22.17 0.8746
## beta_pc_non_0 bf
## 1 0.1254 6.974
Model Comparison - BASE Data II
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 2 10000 0.8 1.75 0.1 0.08 10 0.49 1.8 1.87 10.31 61.59
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 1770 16.84 484 34.45 990 2619
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 255 91.13 2618 1888 58.1 0.7126
## beta_pc_non_0 bf
## 1 0.2874 2.479
Gamma Prior \(\Gamma(\beta; 3, 3)\)
Model Comparison - SSE data I
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 1 10000 0.8 0.54 0.1 0.08 10 9 1.8 2.18 79.71 89.08
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 3849 83.89 3625 87.06 3762 1285
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 3036 29.74 1285 352 78.5 0.5678
## beta_pc_non_0 bf
## 1 0.4322 1.314
Model Comparison - SSE data II
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 1 10000 0.8 0.63 0.1 0.09 10 46.62 1.8 3.41 36.21 31.74
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 3174 61.09 6108 53.19 5318 0
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 9999 0 0 0 NaN 0
## beta_pc_non_0 bf
## 1 1 0
Model Comparison - SSE data III
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 2 10000 0.8 0.42 0.1 0.19 10 10.81 1.8 2.32 15.31 9.4
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 940 9.34 934 13.58 1358 0
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 9999 0 0 0 NaN 0
## beta_pc_non_0 bf
## 1 1 0
Model Comparison - SSE data IV
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 3 10000 0.8 0.95 0.1 0.05 10 0.82 1.8 1.06 17.04 82.39
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 2045 48.23 1197 81.91 2033 1766
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 716 71.15 1765 1398 55.8 0.7518
## beta_pc_non_0 bf
## 1 0.2482 3.029
Model Comparison - BASE Data I
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 1 10000 0.8 1.77 0.1 0.02 10 0.19 1.8 1.79 3.74 53.91
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 676 17.38 218 36.44 457 1238
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 16 98.72 1237 4343 22.17 0.8746
## beta_pc_non_0 bf
## 1 0.1254 6.974
Model Comparison - BASE Data II
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 2 10000 0.8 1.75 0.1 0.08 10 0.49 1.8 1.87 10.31 61.59
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 1770 16.84 484 34.45 990 2619
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 255 91.13 2618 1888 58.1 0.7126
## beta_pc_non_0 bf
## 1 0.2874 2.479
Gamma Prior \(\Gamma(\beta; 4, 4)\)
Model Comparison - SSE data I
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 1 10000 0.8 0.54 0.1 0.09 10 13.97 1.8 3.08 79.43 87.93
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 3862 85.93 3774 85.77 3767 1174
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 3218 26.73 1174 284 80.52 0.5607
## beta_pc_non_0 bf
## 1 0.4393 1.276
Model Comparison - SSE data II
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 1 10000 0.8 0.67 0.1 0.07 10 115.87 1.8 5.09 36.44 30.25
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 3025 65.45 6544 58.15 5814 0
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 9999 0 0 0 NaN 0
## beta_pc_non_0 bf
## 1 1 0
Model Comparison - SSE data III
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 2 10000 0.8 0.4 0.1 0.19 10 10.5 1.8 2.27 16.22 9.85
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 985 8.77 877 14.09 1409 0
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 9999 0 0 0 NaN 0
## beta_pc_non_0 bf
## 1 1 0
Model Comparison - SSE data IV
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 3 10000 0.8 0.9 0.1 0.04 10 0.85 1.8 1.02 16.68 84.78
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 2005 49.73 1176 82.88 1960 1716
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 649 72.56 1715 1414 54.81 0.7635
## beta_pc_non_0 bf
## 1 0.2365 3.228
Model Comparison - BASE Data I
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 1 10000 0.8 1.77 0.1 0.02 10 0.19 1.8 1.79 3.75 54.83
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 687 17.72 222 36.39 456 1238
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 15 98.8 1238 4308 22.32 0.8746
## beta_pc_non_0 bf
## 1 0.1254 6.974
Model Comparison - BASE Data II
## rep n_mcmc alpha a_mc beta b_mc gamma g_mc R0 R0_mc accept_rate_a a_rte_b
## 1 2 10000 0.8 1.75 0.1 0.09 10 0.54 1.8 1.9 10.1 59.74
## n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1 1751 16.41 481 32.82 962 2596
## n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1 335 88.57 2595 1889 57.87 0.7069
## beta_pc_non_0 bf
## 1 0.2931 2.412
Bayes Factor - Summary of Results
In this particular setting in which the Baseline model \(\alpha\) and SSE model \((\alpha, \beta, \gamma)\) are compared, the Bayes factor is calculated as;
\[ \dfrac{Proportion \hspace{1 mm}of \hspace{1 mm} \beta \hspace{1 mm} mcmc \hspace{1 mm} samples == 0}{Proportion \hspace{1 mm}of \hspace{1 mm} \beta \hspace{1 mm} mcmc \hspace{1 mm} samples != 0} \]
The following table summaries the results of the RJMCMC iterations for a number of datasets when both a \(exp(\beta, 1)\) prior and a variety of \(\Gamma(\beta;)\) priors on beta were used.
| Epidemic Data | Max daily infection count | Bayes Factor exp(beta; 1) | Bayes Factor,gamma(2, 2.5) | Bayes Factor,gamma(3, 2) | Bayes Factor,gamma(3, 3) | Bayes Factor,gamma(4, 4) |
|---|---|---|---|---|---|---|
| SSE - dies out | 2 | 7.818 | 0.009 | 1.314 | 1.314 | 1.276 |
| SSE spreads | 55 | 0 | 0 | 0 | 0 | 0 |
| SS spreads | 340 | 0 | 0 | 0 | 0 | 0 |
| SSE - dies out | 2 | 5.481 | 2.052 | 3.029 | 3.029 | 3.228 |
| Base - spreads | 65 | 6.968 | 7.013 | 6.974 | 6.974 | 6.974 |
| Base - spreads | 45 | 2.609 | 2.454 | 2.479 | 2.479 | 2.412 |